Finite difference Hartree-Fock method development

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

An overview of the development of the finite difference Hartree-Fock method for diatomic molecules has been presented. Given the power of the currently available workstations the exact solutions of the HF equations for systems containing 20-25 electrons can be obtained routinely. The approach has been successfully used to calibrate even-tempered gaussian basis sets the discrepancies between the finite difference and the finite basis set results for a number of diatomic molecules (including a 40-electron system GaF) could be narrowed to a couple of iHartree. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

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